CTET Paper II (Mathematics and Science) Sep 2015 Solved Paper

Concept:The polynomial can be rewritten as a quadratic expression in the variable $(2x - y)$. This substitution simplifies the factorization.Explanation:First, rearrange the terms: $4x^{2} + y^{2} - 4xy + 14x - 7y + 12$.Notice that $4x^{2} + y^{2} - 4xy = (2x - y)^{2}$.Also, $14x - 7y = 7(2x - y)$.So the expression becomes $(2x - y)^{2} + 7(2x - y) + 12$.Let $a = 2x - y$. Then we have $a^{2} + 7a + 12$.Factor this quadratic: $a^{2} + 4a + 3a + 12 = (a+4)(a+3)$.Substitute back $a = 2x - y$: $(2x - y + 4)(2x - y + 3)$.Thus, the two factors are $2x - y + 4$ and $2x - y + 3$.Among the given options, $2x - y + 3$ matches option C.Answer:Option C: $2x - y + 3$